Lesson 1

In this section, we will see how to calculate probabilities recursively. To do this, we will consider the simplest version of a gambling game and calculate the probability of gambler's ruin. The theory of probability arose from the analysis of gambling games: the questions of predicting wins were posed by Gerolamo Cardano, Blaise Pascal, Pierre Fermat, and Christiaan Huygens.

Let \(n\) denote the initial capital of the player, who makes a series of bets. If the bet wins, the player's capital increases by one unit, and if it loses, it decreases by one unit. Let \(p\) be the probability of a win, and \(q=1-p\) be the probability of a loss. When \(p=q=\frac{1}{2}\), the game is called unbiased or symmetrical, otherwise it is called biased or asymmetrical.

An abstract model of such a game is a random walk on a line. Initially, the particle is at point \(n\). With probability \(p\), the particle moves one step to the right, and with the remaining probability \(q=(1-p)\), it moves one step to the left. Then, the walk continues.

In the original gambling game, there is a natural lower bound on the capital: when the capital becomes zero, the game ends. This event is called gambler's ruin. There may also be an upper bound: if the player's goal is to increase the capital to \(t\), then the amount \(t-n\) is called the intended profit. In this situation, the particle wanders in the corridor \([0, t]\). As soon as the particle reaches one of the walls of the corridor, the walk ends. The trajectory of the particle can be conveniently visualized as follows.